View source for Risk difference

[[File:Illustration of risk reduction.svg|alt=Illustration of two groups: one exposed to a risk factor, and one unexposed. Exposed group has smaller risk of adverse outcome (RD = −0.25, ARR = 0.25).|thumb|The adverse outcome (black) risk difference between the group exposed to the treatment (left) and the group unexposed to the treatment (right) is −0.25 (RD = −0.25, ARR = 0.25).]]
The '''risk difference''' (RD), '''excess risk''', or '''attributable risk'''<ref name=":02">{{cite book|url=http://www.oxfordreference.com/view/10.1093/acref/9780199976720.001.0001/acref-9780199976720|title=Dictionary of Epidemiology|year=2014 |publisher=Oxford University Press|isbn=978-0-19-939006-9|veditors=Porta M|edition=6th|pages=14|doi=10.1093/acref/9780199976720.001.0001}}</ref> is the difference between the [[risk]] of an outcome in the exposed group and the unexposed group. It is computed as <math>I_e - I_u</math>, where <math>I_e</math> is the [[Incidence_(epidemiology) | incidence]] in the exposed group, and <math>I_u</math> is the incidence in the unexposed group. If the risk of an outcome is increased by the exposure, the term '''absolute risk increase''' (ARI) is used, and computed as <math>I_e - I_u</math>. Equivalently, if the risk of an outcome is decreased by the exposure, the term '''absolute risk reduction''' (ARR) is used, and computed as <math>I_u - I_e</math>.<ref name=":0">{{Cite web|url=http://www.oxfordreference.com/view/10.1093/acref/9780199976720.001.0001/acref-9780199976720|title=Dictionary of Epidemiology - Oxford Reference|year=2014|publisher=Oxford University Press |language=en|doi=10.1093/acref/9780199976720.001.0001|isbn=9780199976720|access-date=2018-05-09|editor1-last=Porta|editor1-first=Miquel}}</ref><ref name=":1">{{Cite book|title=Epidemiology : an introduction|last=J.|first=Rothman, Kenneth|date=2012|publisher=Oxford University Press|isbn=9780199754557|edition=2nd|location=New York, NY|pages=66, 160, 167|oclc=750986180}}</ref>

The inverse of the absolute risk reduction is the [[number needed to treat]], and the inverse of the absolute risk increase is the [[number needed to harm]].<ref name=":0" />

==Usage in reporting==
It is recommended to use absolute measurements, such as risk difference, alongside the relative measurements, when presenting the results of randomized controlled trials.<ref>{{cite journal|vauthors=Moher D, Hopewell S, Schulz KF, Montori V, Gøtzsche PC, Devereaux PJ, Elbourne D, Egger M, Altman DG|date=March 2010|title=CONSORT 2010 explanation and elaboration: updated guidelines for reporting parallel group randomised trials|journal=BMJ|volume=340|pages=c869|doi=10.1136/bmj.c869|pmc=2844943|pmid=20332511}}</ref> Their utility can be illustrated by the following example of a hypothetical drug which reduces the risk of [[colon cancer]] from 1 case in 5000 to 1 case in 10,000 over one year. The relative risk reduction is 0.5 (50%), while the absolute risk reduction is 0.0001 (0.01%). The absolute risk reduction reflects the low probability of getting colon cancer in the first place, while reporting only relative risk reduction, would run into risk of readers exaggerating the effectiveness of the drug.<ref>{{cite journal|last1=Stegenga|first1=Jacob|date=2015|title=Measuring Effectiveness|url=https://www.academia.edu/16420844|journal=Studies in History and Philosophy of Biological and Biomedical Sciences|volume=54|pages=62–71|doi=10.1016/j.shpsc.2015.06.003|pmid=26199055}}</ref>

Authors such as [[Ben Goldacre]] believe that the risk difference is best presented as a [[natural number]] - drug reduces 2 cases of colon cancer to 1 case if you treat 10,000 people. Natural numbers, which are used in the number needed to treat approach, are easily understood by non-experts.<ref name="isbn0-00-724019-8">{{cite book|title=Bad Science|author=Ben Goldacre|publisher=Fourth Estate|year=2008|isbn=978-0-00-724019-7|location=New York|pages=239–260}}</ref>

== Inference ==

Risk difference can be estimated from a 2x2 [[contingency table]]:

{| class="wikitable"
! rowspan="2" |&nbsp;
! colspan="2" |Group
|-
!Experimental (E)
! [[Scientific control|Control]] (C)
|-
| Events (E)
| EE
| CE
|-
| Non-events (N)
| EN
| CN
|}

The [[point estimate]] of the risk difference is

: <math>RD = \frac{EE}{EE + EN} - \frac{CE}{CE + CN}.</math>

The [[sampling distribution]] of RD is approximately [[normal distribution|normal]], with [[standard error]]

: <math>SE(RD) = \sqrt{\frac{EE\cdot EN}{(EE + EN)^3} + \frac{CE\cdot CN}{(CE + CN)^3}}.</math>

The <math>1 - \alpha</math> [[confidence interval]] for the RD is then

: <math>CI_{1 - \alpha}(RD) = RD\pm SE(RD)\cdot z_\alpha,</math>

where <math>z_\alpha</math> is the [[standard score]] for the chosen level of [[Statistical significance|significance]]<ref name=":1" />

== Bayesian interpretation ==

We could assume a disease noted by <math>D</math>, and no disease noted by <math>\neg D</math>, exposure noted by <math>E</math>, and no exposure noted by <math>\neg E</math>. The risk difference can be written as

:<math>RD = P(D\mid E)-P(D\mid \neg E).</math>

==Numerical examples==

=== Risk reduction ===
{{RCT risk reduction example}}

=== Risk increase ===
{{RCT risk increase example}}

==See also==
*[[Population Impact Measures]]
*[[Relative risk reduction]]

==References==
{{reflist}}

{{Medical research studies}}
{{Authority control}}

[[Category:Epidemiology]]
[[Category:Medical statistics]]